Constraints on ultra-low-frequency gravitational waves from an eccentric supermassive black hole binary

Millisecond pulsars with highly stable periods can be considered as very precise clocks and can be used for pulsar timing arrays (PTAs), which attempt to detect nanohertz gravitational waves (GWs) directly. The main sources of nanohertz GWs are supermassive black hole (SMBH) binaries with sub-parsec-scale orbits. On the other hand, an SMBH binary in an earlier phase with a parsec-scale orbit emits ultra-low-frequency (less than or similar to 10(-9) Hz) GWs and cannot be detected with the conventional PTA methodology. Such binaries tend to attain high eccentricity, possibly similar to 0.9. In this paper, we develop a formalism for extending the constraints on GW amplitudes from single sources obtained by PTAs toward ultra-low frequencies considering the waveform expected from an eccentric SMBH binary. GWs from eccentric binaries are contributed from higher harmonics and therefore have a different waveform to those from circular binaries. Furthermore, we apply our formalism to several hypothetical SMBH binaries at the centre of nearby galaxies, including M87, using the constraints from NANOGrav's 11-yr data set. For a hypothetical SMBH binary at the centre of M87, the typical upper limit on the mass ratio is 0.16 for an eccentricity of 0.9 and a semimajor axis of a = 1 pc, assuming the binary phase to be the pericentre.